Virtual COMSATS Inferential Statistics Lecture18 Ossam Chohan Assistant

Virtual COMSATS Inferential Statistics Lecture-18 Ossam Chohan Assistant Professor CIIT Abbottabad 1

Recap of previous lectures • We started hypothesis testing: – Directional hypothesis. – One tail and two tailed Hypothesis. – Specification of Hypothesis. – Level of Significance. – Test Statistics. – Critical Region and conclusions. 2

Objective of lecture-18 • Introduction to Hypothesis testing. – Critical Region Review. – Some Problems 3

Critical Region Review 4

Cont… 5

Assessment Problem-3 • We know that the statement • P( -1. 96δ <µ< +1. 96δ )=0. 95 is correct, while the statement P(140<µ<160)=0. 95 is not correct. Explain why the latter is erroneous? • Some guidelines…. 6

Z values for various confidence levels • • • 99. 73% 99% 98% 96% 95. 45% 90% 80% 68. 27% 50% 3. 00 2. 58 2. 33 2. 05 2. 00 1. 96 1. 645 1. 28 1. 00 0. 6745 7

Problem-1 • The mean lifetime of electric light bulbs produced by a company has in the past been 1120 hours with standard deviation of 125 hours. A sample of 100 electric bulbs recently chosen from a supply of newly produced bulbs showed a mean lifetime of 1070 hours. Test the hypothesis that the mean lifetime of bulbs has not changed, using 5% levels of significance. 8

Problem-1 Solution 9

Problem-2 • The mean weight of a tablet of a certain drug is claimed to be 50 mg. A sample of 100 tablets showed a mean weight of 50. 15 mg with a standard deviation of 0. 4 mg. Using a 1% level of significance, can we conclude that the desired weight is not properly maintained? 10

Problem-2 Solution 11

Assessment problem-1 • A manufacturer supplies the rear axles for U. S Postal Service mail trucks. These axles must be able to withstand 80, 000 pounds per square inch in stress tests, but an excessively strong axle raises production costs significantly. Long experience indicates that the standard deviation of the strength of its axles is 4, 000 pounds per square inch. The manufacturer selects a sample of 100 axles from production, tests them, and finds that the mean stress capacity of the sample is 79, 600 pounds per square inch. 12

Assessment Problem-2 • It has been found from experience at the mean breaking strength of a particular brand of threads is 9. 63 N with a standard deviation of 1. 40 N. Recently a sample of 36 pieces of threads showed a mean breaking strength of 8. 93 N. Can we conclude at 5% and 1% levels of significance that the threads have become inferior? 13

Problem-3 • A company claims that the average lifetime of his product is 2000 hours. A random sample of 64 products is put on test and their lifetime in hours is recorded. The following sums are obtained from the lifetimes: Σx = 127808 and Σ(x- )2 = 9694. 6. Test the hypothesis that the manufacturer is overestimating the lifetimes of the products. Take α = 0. 01. • 14

Problem-3 Solution 15

Problem-4 • Individual filing of income tax returns prior to 30 th June had an average refund of Rs. 1200. Consider the population of last minutes filers who file their returns during the last week of June. For a random sample of 400 individuals who filed a return between 25 and 30 June, the sample mean refund was Rs. 1054 and the sample standard deviation was Rs. 1600. Using 5% level of significance, test the belief that the individuals who wait until the last week of June to file their returns to get a higher refund than early the filers. 16

Problem-4 Solution 17

Assessment Problem-3 • A package device is set to fill detergent powder packets with a mean weight of 5 kg, with standard deviation of 0. 21 kg. The weight of packets can be assumed to be normally distributed. The weight of packets is known to drift upwards over a period of time due to machine fault, which is not tolerable. A random sample of 100 packets is taken and weighted. This sample has a mean weight of 5. 03 kg. Can we conclude that the mean weight produced by the machine has increased? Use a 5% level of significance. 18

Assessment Problem-4 • The mean life span of a sample fluorescent LEDs produced by a company is found to be 1600 days with a standard deviation of 150 days. Test the hypothesis that the mean life span of fluorescent LEDs produced in general is higher than the mean life of 1570 days at α = 0. 01 level of significance. 19

t-distribution table 20

Problem-5 • Researchers are interested in whether the mean level of enzyme B in a certain population is different from 120. They measure levels of enzyme B in a sample of 15 individuals and find that the mean, = 96 and the sample standard deviation, s = 35. 21

Problem-5 Solution 22

Problem-6 • The Average breaking strength of steel rods is specified to be 18. 5 thousand kg. For this a sample of 14 rods was tested. The mean and standard deviation obtained were 17. 85 and 1. 955, respectively. Test the significance of the deviation. 23

Problem-6 Solution 24

Assessment Problem-5 • An automobile tyre manufacturer claims that the average life of a particular grade of tyre is more than 20, 000 km when used under normal conditions. A random sample of 16 tyres was tested and a mean and standard deviation of 22, 000 km and 5, 000 km, respectively were computed. Assuming the life of the tyres in km to be approximately normally distributed, decide whether the manufacturer’s claim is valid. 25

Assessment Problem-6 • A random sample of 22 fifth grade pupils have a grade point average of 5. 0 in maths with a standard deviation of 0. 452, whereas marks range from 1 (worst) to 6 (excellent). The grade point average (GPA) of all fifth grade pupils of the last five years is 4. 7. Is the GPA of the 22 pupils different from the populations’ GPA? 26

Conti… Pupil Grade Point 1 5 13 5. 5 2 5. 5 14 5. 5 3 5. 5 15 4 4 5 16 5 5 5 17 5 6 6 18 5. 5 7 5 19 4. 5 8 5 20 5. 5 9 4. 5 21 5 10 5 22 5. 5 11 5 Mean 5. 0 12 4. 5 Variance 0. 2045 27